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Graphing Quadratic Inequalities

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Graphing Quadratic Inequalities

Study Guide

Key Definition

A quadratic inequality of the form $y > ax^2 + bx + c$ represents the unbounded region above the parabola $y = ax^2 + bx + c$.

Important Notes

If the inequality is $\\\1$ or $\\\1$, use a solid curve because the boundary is included.
If the inequality is $<$ or $>$, use a dashed curve because the boundary is not included.
The vertex of the parabola is at $(-\\\1{b},{2a},\, c - \\\1{b^2},{4a})$ or simply use $x = -\\\1{b},{2a}$ for the x-coordinate.
The parabola opens upward if $a > 0$ and downward if $a < 0$.
To decide which side to shade, pick a test point not on the parabola and check the inequality.

Mathematical Notation

$a$: coefficient of $x^2$, controls the opening direction

$b$: coefficient of $x$

$c$: constant term, shifts the parabola vertically

$x = -\\\1{b},{2a}$: x-coordinate of the vertex

$\\\1 = b^2 - 4ac$: discriminant, indicates number of x-intercepts

Use these symbols to identify and graph the parabola before shading the solution region.

Why It Works

When graphing an inequality like $y \\\1 ax^2 + bx + c$, first determine if the boundary is included: use a solid curve for “≥/≤” and dashed for “>/ <.” Next, sketch the parabola by finding its vertex and intercepts. Finally, choose a test point not on the curve to see which side satisfies the inequality, and shade that region.

Remember

Always determine the direction of the parabola from the sign of $a$.

Quick Reference

Vertex Formula:$x = -\\\1{b},{2a}$

Factored Form:$y = (x + p)(x + q)$

Understanding Graphing Quadratic Inequalities

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

To graph $y \\\1 x^2 - x - 12$, sketch the parabola $y = x^2 - x - 12$ by finding its vertex and x-intercepts. Draw a solid curve because “≤” includes the boundary. Pick a test point not on the curve (for example, (0,0)), check the inequality, and then shade the region that makes the inequality true (the area below the parabola).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which inequality represents the region below the parabola $y = x^2 - x - 12$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboard ramp is represented by the parabola $y = -x^2 + 6x - 8$. Determine the region above the ramp.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Find the vertex and axis of symmetry for the parabola $y = x^2 - 4x + 3$.

Challenge Quiz

Single Choice Quiz

Advanced

For the inequality $y \\\1 2x^2 - 3x + 1$, which statement is true?

Recap

Watch & Learn

Review key concepts and takeaways